Discontinuous Galerkin Methods
In this final chapter we present the discontinuous Galerkin (dG) method. This method is based on finite element spaces that consist of discontinuous piecewise polynomials defined on a partition of the computational domain. Such methods are very flexible,
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Discontinuous Galerkin Methods
Abstract In this final chapter we present the discontinuous Galerkin (dG) method. This method is based on finite element spaces that consist of discontinuous piecewise polynomials defined on a partition of the computational domain. Such methods are very flexible, for example, since they allow construction of more general methods and since they allow for simple adaptation. Discontinuous Galerkin methods were originally developed for first order problems and were later extended to second order problems. We cover both categories, and derive basic stability and error estimates. Due to the discontinuous nature of the finite element space additional terms in the weak form are necessary to enforce the proper continuity conditions between adjacent elements. We also consider how to handle these additional terms in the implementation of the method.
14.1 A First Order Problem 14.1.1 Model Problem Let ˝ be a domain in Rd , d D 1; 2, or 3 with boundary @˝. Let b D Œbi diD1 be a given vector field and c a given scalar function. We consider the following first order problem modeling convection and reaction: find u such that cu C b ru D f; u D g;
in ˝
(14.1a)
on @˝
(14.1b)
where @˝ D fx 2 @˝ W n.x/ b.x/ < 0g
(14.2)
is the so-called inflow part of the boundary. For simplicity, we assume that b is a constant vector and c a constant function. M.G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Texts in Computational Science and Engineering 10, DOI 10.1007/978-3-642-33287-6__14, © Springer-Verlag Berlin Heidelberg 2013
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14 Discontinuous Galerkin Methods
Fig. 14.1 The inflow boundary @˝ , where n b < 0
The operators c and b r can be simply interpreted. The first scales u so that it is proportional to f . The second, transports u along the direction of b. To describe this we speak about reaction and convection. Note that there is no diffusion term like u, which can smooth u and make it adhere to a boundary condition. Therefore, we can only have boundary conditions on the inflow part of the boundary. That is, the parts of @˝ where the vectors of b point into ˝. See Fig. 14.1.
14.1.2 Discontinuous Finite Element Spaces Let K D fKg be a mesh of ˝ and define the space of discontinuous piecewise linear functions Vh D fv W vjK 2 P1 .K/; 8K 2 Kg
(14.3)
where P1 .K/ is the space of linear polynomials on element K. Thus, the members of Vh are linear on each element K, but generally discontinuous across the element boundaries @K. As before, we let EI denote the set of interior edges and with each interior edge E we associate a fixed unit normal n. We denote by K C the element for which n is the exterior normal, and K the element for which n is the exterior normal. For edges on the boundary @˝ we let n be the exterior unit normal to ˝. Further, we define the jump and the average of a function v 2 Vh at the edge E by
14.1 A First Order Problem
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Œv D vC v ;
hvi D
uC C u 2
(14.4)
14.1.3 The Discontinuous Galerkin Meth
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